6. Tabu Search6.4 Tabu Search for TSP

Fall 2010

Instructor: Dr. Masoud Yaghini

Tabu Search: Part 4

Outline

� Basic TS Components

� Advanced Strategies

� References

Basic TS Components

Tabu Search: Part 4

Solution Representation

� A solution can be represented by a vector indicating

the order the cities are visited

– Example:

1 5 3 4 2 6 7 8

– Numbers in the solution vector are interpreted as cities

and not as positions in the solution vector

Tabu Search: Part 4

Initial Solutions

� A good feasible, yet not-optimal, solution to the

TSP can be found quickly using a greedy approach

(the nearest-neighbor heuristic).

� Starting with the first node in the tour, find the

nearest node. nearest node.

� Each time find the nearest unvisited node from the

current node until all the nodes are visited.

Tabu Search: Part 4

Neighborhood Structure

� A neighborhood to a given solution is defined as any other

solution that is obtained by a pair wise exchange of any two

nodes in the solution.

� This always guarantees that any neighborhood to a feasible

solution is always a feasible solution (i.e, does not form any

sub-tour). sub-tour).

� If we fix node 1 as the start and the end node, for a problem

of N nodes, there are CN2 such neighborhoods to a given

solution.

� At each iteration, the neighborhood with the best objective

value (minimum distance) is selected.

� Neighborhood cardinality:

Tabu Search: Part 4

Neighborhood Structure

� Neighborhood solution obtained by swapping the

order of visit of cities 2 and 5

1 2 3 4 5 6 7 8 1 5 3 4 2 6 7 8

Tabu Search: Part 4

Tabu List

� To prevent the process from cycling in a small set

of solutions, some attribute of recently visited

solutions is stored in a Tabu List, which prevents

their occurrence for a limited period.

� The attribute used is a pair of nodes that have been � The attribute used is a pair of nodes that have been

exchanged recently.

� A Tabu structure stores the number of iterations for

which a given pair of nodes is prohibited from

exchange.

Tabu Search: Part 4

Tabu List

� The main memory component (short term memory) can be

stored in a matrix where the swap of cities i and j is recorded

in the i-th row and j-th column

� The memory can be organised as upper-triangular matrix

1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8

Tabu Search: Part 4

Short term memory

� An example of Short Term Memory (M)Most

Recent

Swap

1 2 3 4 5 6 7 8

1 1

2 5

3 4

4 3

5 2

6

7

8

the number of

remaining iterations

that given swap

stays on the Tabu

list

Tabu Search: Part 4

Aspiration Criterion

� Tabus may sometimes prohibit attractive moves.

� It may, therefore, become necessary to revoke

tabus at times.

� The criterion used is to allow a move, even if it is

tabu, if it results in a solution with an objective tabu, if it results in a solution with an objective

value better than that of the current best-known

solution.

Tabu Search: Part 4

Termination Criteria

� The algorithm terminates if there is no improvement

in the solution for a pre-defined number of

iterations

Advanced Strategies

Tabu Search: Part 4

A Speedup Technique

� For larger TSP problems if a small value ranging

between 15 and 40 cities of the nearest-neighbor list

is constructed, all local search can be done only on

this nearest-neighbor list

� For example when we want to swap two cities, � For example when we want to swap two cities,

– First we choose a city and then

– Check swapping only with the nearest-neighbor list

Tabu Search: Part 4

Diversification

� Quite often, the process may get trapped in a space

of local optimum.

� To allow the process to search other parts of the

solution space (to look for the global optimum), it is

required to diversify the search process, driving it required to diversify the search process, driving it

into new regions.

� This is implemented using frequency based

memory.

Tabu Search: Part 4

Diversification

� The use of frequency information is used to penalize non-

improving moves by assigning a larger penalty (frequency

count adjusted by a suitable factor) to swaps with greater

frequency counts.

� This diversifying influence is allowed to operate only on

occasions when no improving moves exist. occasions when no improving moves exist.

� Additionally, if there is no improvement in the solution for a

pre-defined number of iterations, frequency information can

be used for a pair wise exchange of nodes that have been

explored for the least number of times in the search space.

� Thus driving the search process to areas that are largely

unexplored so far.

Tabu Search: Part 4

Diversification

� Long term memory:

– Maintain a list of t towns which have been considered in

the last k best (worst) solutions

– encourage (or discourage) their selections in future

solutions solutions

– using their frequency of appearance in the set of elite

solutions and the quality of solutions which they have

appeared in our selection function

Tabu Search: Part 4

Diversification

� An example of Long Term Memory (H) last 50

iterations

2 3 4 5 6 7 8

0 2 3 3 0 1 1 10 2 3 3 0 1 1 1

2 1 3 1 1 0 2

2 3 3 4 0 3

1 1 2 1 4

4 2 1 5

3 1 6

6 7

Tabu Search: Part 4

Other Local Search Methods

� Swap operation of two cities on the tour for

selecting the neighborhood is not the best choice for

Tabu Search.

� Three other types of local search for the TSP:

– 2-opt neighborhood– 2-opt neighborhood

– 2.5-opt neighborhood

– 3-opt neighborhood

Tabu Search: Part 4

2-opt neighborhood

� The 2–opt neighborhoods in the TSP

� Given a candidate solution s, the TSP 2–opt neighborhood

of a candidate solution s consists of the set of all the

candidate solutions s’ that can be obtained from s by

exchanging two pairs of arcs in all the possible ways.

Example: the pair of arcs (b, c) and (a, f) is removed and � Example: the pair of arcs (b, c) and (a, f) is removed and

replaced by the pair (a, c) and (b, f)

Tabu Search: Part 4

3-opt neighborhood

� The 3-opt neighborhood consists of those tours that can be

obtained from a tour s by replacing at most three of its arcs.

� In a 3-opt local search procedure 2-opt moves are also

examined. Example:

Tabu Search: Part 4

2.5-opt neighborhood

� 2.5-opt checks whether inserting the city between a

city i and its successor, as illustrated in the figure

below, results in an improved tour.

The End